A month left to do what, that's the question. Ifyou have works, staying at the big cities like Hanoi or HCMC is a must, obviously.
But if you have that month to play around... now that is interesting.
Hue is renown in Vietnam as a cheap tourist city. I mean, a cheap city for tourists to live. Cheap accomodation, cheap (but great) food even to Vietnamese. A month to explore every crooks and cranies of that ex-capital can be a good time. If I have spare time to tour Vietnam, I would choose Hue. The pace of life is slow, so it might not to your taste.
Da Nang lately is praised to be an easy city. But I dont know. For a visitor, Da Nang may not have enough to offer for one month long.
I will leave HCMC to her fans here to praise.
But how about Hanoi? Live in the central district can be quite expensive, but you can choose a place far from it. Long Bien Chuong Duong, on the north east, over the old bridges are plentiful of cheap hotels and it's nearer, faster to go from there to Hoan Kiem district with its old quarter.
Consequently, move to Ha Dong area, far from HK, but near many many old villages. Forewarn: move from Ha Dong back to Old Quarter can be quite long because all the roads are crowded, even major roads. LB-CD is faster because it's only crowded starting from the bridges.
The new districts on the west Tu Liem, etc.. are full of commercial and office towers, so it might be cheap, but where do you go from there? Eastward to Old Quarter is slowed.